Umbilic point
In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are equal, and every tangent vector is a principal direction. The name "umbilic" comes from the Latin umbilicus - navel.
Umbilic points generally occur as isolated points in the elliptical region of the surface; that is, where the Gaussian curvature is positive. For surfaces with genus 0, e.g. an ellipsoid, there must be at least four umbilics, a consequence of the Poincaré–Hopf theorem. An ellipsoid of revolution has only two umbilics.
The sphere is the only surface with non-zero curvature where every point is umbilic. A flat umbilic is an umbilic with zero Gaussian curvature. The monkey saddle is an example of a surface with a flat umbilic and on the plane every point is a flat umbilic. A torus can have no umbilics, but every closed surface of nonzero Euler characteristic, embedded smoothly into Euclidean space, has at least one umbilic. An unproven conjecture of Constantin Carathéodory states that every smooth topological sphere in Euclidean space has at least two umbilics.
The three main type of umbilic points are elliptical umbilics, parabolic umbilics and hyperbolic umbilics. Elliptical umbilics have the three ridge lines passing through the umbilic and hyperbolic umbilics have just one. Parabolic umbilics are a transitional case with two ridges one of which is singular. Other configurations are possible for transitional cases. These cases correspond to the D4−, D5 and D4+ elementary catastrophes of René Thom's catastrophe theory.
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